State the product rule for differentiating \(f(x)g(x)\).

In the product rule, what are the two terms for \(u(x)v(x)\)?

Differentiate \(x\sin x\).

Differentiate \(x e^x\).

Differentiate \(x^2\sin x\).

Differentiate \((x^3)(e^x)\).

Differentiate \((x^2+1)\cos x\).

Differentiate \((3x-1)\ln x\) for \(x>0\).

Find \(f'(1)\) for \(f(x)=x^2e^x\).

Differentiate \(x^2(x^3+1)\) using the product rule, then simplify.

Differentiate \((x^2+2x)(\sin x+1)\).

Explain why \(\frac{d}{dx}[f(x)g(x)]\) is not generally \(f'(x)g'(x)\).

Differentiate \(x\sin x\cos x\).

Differentiate \((x^2+1)e^x\sin x\).

Find \(a\) if \(f(x)=(ax+1)e^x\) has \(f'(0)=5\).

Find all \(x\) where \(f(x)=x e^x\) has a horizontal tangent.

Find \(k\) if \(f(x)=x^2(kx+1)\) has \(f'(1)=10\).

A student differentiates \(x^2\sin x\) as \(2x\cos x\). Identify the error and correct derivative.

Derive the product rule from the difference quotient in outline form.

When is expanding before differentiating \(x^2(x^3+1)\) equivalent to using the product rule? Explain.